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We show that the formula for the essential norm of a multiplication operator on    $$L_p$$  L p , for    $$1&lt;p&lt;\infty $$  1 &lt; p &lt; ∞ , also holds for    $$p=1$$  p = 1 . We also provide a proof for the formula which works simultaneously for all    $$p\in [1,\infty )$$  p ∈ [ 1 , ∞ ) .

The essential norm of multiplication operators on $$L_p(\mu )$$